Fall Abstracts

Alex Zupan (Texas)

Totally geodesic subgraphs of the pants graph

Abstract:
For a compact surface S, the associated pants graph P(S) consists of vertices corresponding to pants decompositions of S and edges corresponding to elementary moves between pants decompositions. Motivated by the Weil-Petersson geometry of Teichmüller space, Aramayona, Parlier, and Shackleton conjecture that the full subgraph G of P(S) determined by fixing a multicurve is totally geodesic in P(S). We resolve this conjecture in the case that G is a product of Farey graphs. This is joint work with Sam Taylor.

Jayadev Athreya (Illinois)

Gap Distributions and Homogeneous Dynamics

Abstract:
We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics, generalizing earlier work on gaps between Farey Fractions. This works gives some possible notions of `randomness' of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Lelievre, and with Y.Cheung. This talk may also be of interest to number theorists.

Joel Robbin (Wisconsin)

GIT and [math]\mu[/math]-GIT

Many problems in differential geometry can be reduced to solving a PDE of form

[math]
\mu(x)=0
[/math]

where [math]x[/math] ranges over some function space and [math]\mu[/math] is an infinite dimensional analog of the moment map in symplectic geometry.
In Hamiltonian dynamics the moment map was introduced to use a group action to reduce the number of degrees of freedom in the ODE.
It was soon discovered that the moment map could be applied to Geometric Invariant Theory:
if a compact Lie group [math]G[/math] acts on a projective algebraic variety [math]X[/math],
then the complexification [math]G^c[/math] also acts and there is an isomorphism of orbifolds

[math]
X^s/G^c=X//G:=\mu^{-1}(0)/G
[/math]

between the space of orbits of Mumford's stable points and the Marsden-Weinstein quotient.

In September of 2013 Dietmar Salamon, his student Valentina Georgoulas, and I wrote an exposition of (finite dimensional) GIT from the point of view of symplectic geometry.
The theory works for compact Kaehler manifolds, not just projective varieties.
I will describe our paper in this talk; the following Monday Dietmar will give more details in the Geometric Analysis Seminar.

Anton Lukyanenko (Illinois)

Uniformly quasi-regular mappings on sub-Riemannian manifolds

Abstract:
A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. f(z)=z^2. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:
1) Every lens space admits a uniformly QR (UQR) mapping f.
2) Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

Neil Hoffman (Melbourne)

Verified computations for hyperbolic 3-manifolds

Abstract:
Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?

While this question can be answered in the negative if M is known to
be reducible or toroidal, it is often difficult to establish a
certificate of hyperbolicity, and so computer methods have developed
for this purpose. In this talk, I will describe a new method to
establish such a certificate via verified computation and compare the
method to existing techniques.

This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi,
Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.

Khalid Bou-Rabee (Minnesota)

On generalizing a theorem of A. Borel

The proof of the Hausdorff-Banach-Tarski paradox relies on the existence of a nonabelian free group in the group of rotations of [math]\mathbb{R}^3[/math]. To help generalize this paradox, Borel proved the following result on free groups.

Borel’s Theorem (1983): Let [math]F[/math] be a free group of rank two. Let [math]G[/math] be an arbitrary connected semisimple linear algebraic group (i.e., [math]G = \mathrm{SL}_n[/math] where [math]n \geq 2[/math]). If [math]\gamma[/math] is any nontrivial element in [math]F[/math] and [math]V[/math] is any proper subvariety of [math]G(\mathbb{C})[/math], then there exists a homomorphism [math]\phi: F \to G(\mathbb{C})[/math] such that [math]\phi(\gamma) \notin V[/math].

What is the class, [math]\mathcal{L}[/math], of groups that may play the role of [math]F[/math] in Borel’s Theorem? Since the free group of rank two is in [math]\mathcal{L}[/math], it follows that all residually free groups are in [math]\mathcal{L}[/math]. In this talk, we present some methods for determining whether a finitely generated group is in [math]\mathcal{L}[/math]. Using these methods, we give a concrete example of a finitely generated group in [math]\mathcal{L}[/math] that is *not* residually free. After working out a few other examples, we end with a discussion on how this new theory provides an answer to a question of Brueillard, Green, Guralnick, and Tao concerning double word maps. This talk covers joint work with Michael Larsen.

Morris Hirsch (Wisconsin)

Common zeros for Lie algebras of vector fields on real and complex 2-manifolds.

The celebrated Poincare-Hopf theorem states that a vector ﬁeld [math]X[/math] on a manifold
[math]M[/math] has nonempty zero set [math]Z(X)[/math], provided [math]M[/math] is compact with empty boundary and
[math]M[/math] has nonzero Euler characteristic. Surprising little is known about the set of
common zeros of two or more vector ﬁelds, especially when [math]M[/math] is not compact.
One of the few results in this direction is a remarkable theorem of Christian
Bonatti (Bol. Soc. Brasil. Mat. 22 (1992), 215–247), stated below. When [math]Z(X)[/math] is
compact, [math]i(X)[/math] denotes the intersection number of [math]X[/math] with the zero section of the
tangent bundle.

[math]\cdot [/math] Assume [math] dim_{\mathbb{R}(M)} ≤ 4[/math], [math]X[/math] is analytic, [math]Z(X)[/math] is compact and [math]i(X) \neq 0[/math]. Then
every analytic vector ﬁeld commuting with [math]X[/math] has a zero in [math]Z(X)[/math].
In this talk I will discuss the following analog of Bonatti’s theorem. Let [math]\mathfrak{g}[/math] be
a Lie algebra of analytic vector ﬁelds on a real or complex 2-manifold [math]M[/math], and set
[math]Z(g) := \cap_{Y \in \mathfrak{g}} Z(Y)[/math].

• Assume [math]X[/math] is analytic, [math]Z(X)[/math] is compact and [math]i(X) \neq 0[/math]. Let [math]\mathfrak{g}[/math] be generated by
analytic vector ﬁelds [math]Y[/math] on [math]M[/math] such that the vectors [math][X,Y]p[/math] and [math]Xp[/math] are linearly
dependent at all [math]p \in M[/math]. Then [math]Z(\mathfrak{g}) \cap Z(X) \neq \emptyset [/math].
Related results on Lie group actions, and nonanalytic vector ﬁelds, will also be
treated.

Spring Abstracts

Spencer Dowdall (UIUC)

Fibrations and polynomial invariants for free-by-cyclic groups

The beautiful theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology M^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial."

This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G. Specifically, I will introduce a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G. The polynomial invariant additionally provides a wealth of dynamical information about these splittings. This is joint work with Ilya Kapovich and Christopher J. Leininger.

Ioana Suvaina (Vanderbilt)

ALE Ricci flat Kahler surfaces from a Tian-Yau construction approach"

The talk presents an explicit classification of the ALE Ricci flat Kahler surfaces (M,J,g), generalizing
previous classification results of Kronheimer. The manifolds are related to Q-Gorenstein deformations
of quotient singularities of type C^2/G, with G a finite subgroup of U(2).
Using this classification, we show how these metrics can also be obtained by a construction of Tian-Yau.
In particular, we find good compactifications of the underlying complex manifold M.

Jae Choon Cha (POSTECH)

Universal bounds for the Cheeger-Gromov rho-invariants"

Cheeger and Gromov showed that there is a universal bound of their L2 rho-invariants of a fixed smooth closed (4k-1)-manifold, using a deep analytic method. We give a new topological proof of the existence of a universal bound. For 3-manifolds, we give explicit estimates in terms of triangulations, Heegaard splittings, and surgery descriptions. The proof employs interesting ideas including controlled chain homotopy and a geometric reinterpretation of the Atiyah-Hirzebruch bordism spectral sequence. Applications include new results on the complexity of 3-manifolds.

Mustafa Kalafat (Michigan-State and Tunceli)

We show that a compact complex surface which admits a conformally Kahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric. This is joint work with C.Koca.

Matthew Kahle (Ohio)

TBA

Yongqiang Liu

Nearby cycles and Alexander modules of hypersurface complements

For a polynomial transversal at infinity, we show that the Alexander modules of the hypersurface complement can be realized by the nearby cycle complex, and we obtain a divisibility result for the associated Alexander polynomial. As an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by Dimca and Libgober.

Pallavi Dani (LSU)

A finitely generated group can be endowed with a natural metric which
is unique up to coarse isometries, or quasi-isometries. A fundamental
question is to classify finitely generated groups up to
quasi-isometry. I will report on the progress on this question in the
case of right-angled Coxeter groups. In particular I will describe
how topological features of the visual boundary can be used to
classify a family of hyperbolic right-angled Coxeter groups. I will
also discuss the connection with commensurability, an algebraic
property which implies quasi-isometry, but is stronger in general.
This is joint work with Anne Thomas.

Jingzhou Sun (Stony Brook)

"On the Demailly-Semple jet bundles of hypersurfaces in the 3-dimensional complex projective space"

Let X be a smooth hypersurface of degree d in the 3-dimensional complex projective space.
By totally algebraic calculations, we prove that on the third Demailly-Semple jet bundle X_3 of X,
the Demailly-Semple line bundle is big for d not ness than 11,
and that on the fourth Demailly-Semple jet bundle X_4 of X,
the Demailly-Semple line bundle is big for d not ness than 10, improving a recent result of Diverio.